They probably did it to appease the alpha dad's family. I kind of get it, since he already "failed" them by divorcing his exhusband/wife and then, on top of that, he got married to a beta.
I am more irked that they didn't adopted more. But I guess they didn't want to pressure more kids to compete against each other, and they thought their alpha kid would be less stressed as an only child. Little did they know that their kid got competitive against her cousin even though they never asked her to be better than Byul. I find it kind of ironic.

At first I was kinda disappointed they adopted an alpha from orphanage but then I remembered they're rich. They bought a purebred dog instead of adopting one. Of course they'd adopt an alpha. Sorry all you lowly beta orphans. ¯_(ツ)_/¯
Like, I get why he might not be able to adopt an omega (since one parent being an alpha and one being a beta means the alpha might react if an omega child went into heat...
But seriously. Why not adopt a beta? I bet the orphanages are full of beta children.